Formulas and Properties, Their Links and Characteristics

This paper concerns with the study of two pursuit diﬀerential game problems of many pursuers and many evaders on a nonempty closed convex subset of R^n. Throughout the period of the games, players must stay within the given closed convex set. Players’ laws of motion are deﬁned by certain ﬁrst order diﬀerential equations. Control functions of the pursuers and evaders are subject to geometric constraints. Pursuit is said to be completed if the geometric position of each of the evader coincides with that of a pursuer. We proved two theorems each of which is solution to a problem. Suﬃcient conditions for the completion of pursuit are provided in each of the theorems. Moreover, we constructed strategies of the pursuers that ensure completion of pursuit.

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Spectrum of a Differential Operator with Periodic Generalized Potential

Abstract and Applied Analysis ◽

10.1155/2007/74595 ◽

2007 ◽

Vol 2007 ◽

pp. 1-8

Author(s):

Mehmet Sahin ◽

Manaf Dzh. Manafov

Keyword(s):

Differential Operator ◽

Infinite Number ◽

Periodic Potential ◽

Generalized Functions ◽

Order Differential Operator ◽

Nonzero Constant ◽

Classic Case ◽

First Order ◽

Generalized Potential ◽

The Given

We study some spectral problems for a second-order differential operator with periodic potential. Notice that the given potential is a sum of zero- and first-order generalized functions. It is shown that the spectrum of the investigated operator consists of infinite number of gaps whose length limit unlike the classic case tends to nonzero constant in some place and to infinity in other place.

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Error bounds on multivariate Normal approximations for word count statistics

Advances in Applied Probability ◽

10.1017/s0001867800011769 ◽

2002 ◽

Vol 34 (03) ◽

pp. 559-586 ◽

Cited By ~ 1

Author(s):

Haiyan Huang

Keyword(s):

Covariance Matrix ◽

Error Bounds ◽

Multivariate Distribution ◽

Finite Alphabet ◽

Multivariate Normal ◽

Word Count ◽

First Order ◽

Normal Approximations ◽

Necessary And Sufficient ◽

The Given

Given a sequence S and a collection Ω of d words, it is of interest in many applications to characterize the multivariate distribution of the vector of counts U = (N(S,w 1), …, N(S,w d )), where N(S,w) is the number of times a word w ∈ Ω appears in the sequence S. We obtain an explicit bound on the error made when approximating the multivariate distribution of U by the normal distribution, when the underlying sequence is i.i.d. or first-order stationary Markov over a finite alphabet. When the limiting covariance matrix of U is nonsingular, the error bounds decay at rate O ((log n) / √n) in the i.i.d. case and O ((log n)3 / √n) in the Markov case. In order for U to have a nondegenerate covariance matrix, it is necessary and sufficient that the counted word set Ω is not full, that is, that Ω is not the collection of all possible words of some length k over the given finite alphabet. To supply the bounds on the error, we use a version of Stein's method.

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SURFACE-COVERED 1×3 GRATING WITH DIELECTRIC-METAL SLABS UNDER SECOND BRAGG INCIDENCE

Surface Review and Letters ◽

10.1142/s0218625x19502019 ◽

2020 ◽

Vol 27 (09) ◽

pp. 1950201

Author(s):

CHEN FU ◽

BO WANG ◽

WENHUA ZHU ◽

KUNHUA WEN ◽

ZIMING MENG ◽

...

Keyword(s):

Incident Light ◽

Polarized Light ◽

Transverse Magnetic ◽

Second Order ◽

Bragg Angle ◽

Efficiency Ratio ◽

First Order ◽

Rigorous Coupled Wave Analysis ◽

Rigorous Coupled Wave ◽

The Given

This paper designed a novel three-port reflective surface-covered grating with a connecting layer. The grating can be used as a splitter, and the polarized light can be divided into zero order, first order and second order. Through rigorous coupled-wave analysis, the efficiency of the three orders of diffraction light is close to 33% under the condition that the incident light at 1550 nm is incident at the second Bragg angle and the given duty cycle is 0.5. The efficiency and bandwidth of the surface-covered grating are improved compared with that of the surface-relief grating reported in the past. Especially for transverse magnetic polarized light, the beam splitting effect is more uniform, the efficiency ratio of the zeroth order to first order can reach 1.01, and the efficiency ratio of the first order to second order can reach 1.

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Uniqueness issues for evolution equations with density constraints

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202516500445 ◽

2016 ◽

Vol 26 (09) ◽

pp. 1761-1783 ◽

Cited By ~ 3

Author(s):

Simone Di Marino ◽

Alpár Richárd Mészáros

Keyword(s):

Velocity Field ◽

Evolution Equations ◽

Planck Equation ◽

Wasserstein Distance ◽

First Order ◽

Uniqueness Results ◽

Monotonicity Assumption ◽

Crowd Motion ◽

The Given ◽

Uniqueness Of A Solution

In this paper, we present some basic uniqueness results for evolution equations under density constraints. First, we develop a rigorous proof of a well-known result (among specialists) in the case where the spontaneous velocity field satisfies a monotonicity assumption: we prove the uniqueness of a solution for first-order systems modeling crowd motion with hard congestion effects, introduced recently by Maury et al. The monotonicity of the velocity field implies that the [Formula: see text]-Wasserstein distance along two solutions is [Formula: see text]-contractive, which in particular implies uniqueness. In the case of diffusive models, we prove the uniqueness of a solution passing through the dual equation, where we use some well-known parabolic estimates to conclude an [Formula: see text]-contraction property. In this case, by the regularization effect of the nondegenerate diffusion, the result follows even if the given velocity field is only [Formula: see text] as in the standard Fokker–Planck equation.

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An Interpolation Theorem

Bulletin of Symbolic Logic ◽

10.2307/420966 ◽

2000 ◽

Vol 6 (4) ◽

pp. 447-462 ◽

Cited By ~ 13

Author(s):

Martin Otto

Keyword(s):

Interpolation Theorem ◽

Order Logic ◽

First Order Logic ◽

Elementary Model ◽

Relation Symbol ◽

First Order ◽

Theoretic Proof ◽

Universal Quantification ◽

The Given ◽

Interpolation Result

AbstractLyndon's Interpolation Theorem asserts that for any valid implication between two purely relational sentences of first-order logic, there is an interpolant in which each relation symbol appears positively (negatively) only if it appears positively (negatively) in both the antecedent and the succedent of the given implication. We prove a similar, more general interpolation result with the additional requirement that, for some fixed tuple of unary predicates U, all formulae under consideration have all quantifiers explicitly relativised to one of the U. Under this stipulation, existential (universal) quantification over U contributes a positive (negative) occurrence of U.It is shown how this single new interpolation theorem, obtained by a canonical and rather elementary model theoretic proof, unifies a number of related results: the classical characterisation theorems concerning extensions (substructures) with those concerning monotonicity, as well as a many-sorted interpolation theorem focusing on positive vs. negative occurrences of predicates and on existentially vs. universally quantified sorts.

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Topical Neural Theorem Prover that Induces Rules

10.29007/wscr ◽

2020 ◽

Author(s):

Shuang Xia ◽

Krysia Broda ◽

Alessandra Russo

Keyword(s):

Knowledge Base ◽

Predictive Accuracy ◽

Theorem Prover ◽

Computational Time ◽

Backward Chaining ◽

First Order ◽

Computation Tree ◽

Computation Trees ◽

Reasoning Mechanism ◽

The Given

Various sub-symbolic approaches for reasoning and learning have been proposed in the literature. Among these approaches, the neural theorem prover (NTP) approach uses a backward chaining reasoning mechanism to guide a machine learning architecture to learn vector embedding representations of predicates and to induce first-order clauses from a given knowledge base. NTP is however known for being not scalable, as the computation trees generated by the backward chaining process can grow exponentially with the size of the given knowledge base. In this paper we address this limitation by extending the NTP approach with a topic-based method for controlling the induction of first-order clauses. Our proposed approach, called TNTP for Topical NTP, identifies topic-based clusters over a large knowledge-base and uses these clusters to control the soft unification of predicates during the learning process with the effect of reducing the size of the computation tree needed to induce first-order clauses. Our TNTP framework is capable of learning a diverse set of induced rules that have improved predictive accuracy, whilst reducing the computational time by several orders of magnitude. We demonstrated this by evaluating our approach on three different datasets (UMLS, Kinship and Nations) and comparing our results with that of the NTP method, chosen here as our baseline.

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Breeding Theorem Proving Heuristics with Genetic Algorithms

10.29007/gms9 ◽

2018 ◽

Author(s):

Simon Schäfer ◽

Stephan Schulz

Keyword(s):

Genetic Algorithms ◽

Large Scale ◽

First Order ◽

Proof Search ◽

Theorem Provers ◽

Search Heuristics ◽

Practical Performance ◽

Set Up ◽

The Given ◽

Selection Of

First-order theorem provers have to search for proofs in an infinitespace of possible derivations. Proof search heuristics play a vitalrole for the practical performance of these systems. In the currentgeneration of saturation-based theorem provers like SPASS, E,Vampire or Prover~9, one of the most important decisions is theselection of the next clause to process with the given clausealgorithms. Provers offer a wide variety of basic clause evaluationfunctions, which can often be parameterized and combined in manydifferent ways. Finding good strategies is usually left to the usersor developers, often backed by large-scale experimentalevaluations. We describe a way to automatize this process usinggenetic algorithms, evaluating a population of different strategieson a test set, and applying mutation and crossover operators to goodstrategies to create the next generation. We describe the design andexperimental set-up, and report on first promising results.

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Evaluation of a Two-Dimensional Conductivity Function Based on Boundary Measurements

Journal of Heat Transfer ◽

10.1115/1.521473 ◽

1999 ◽

Vol 122 (2) ◽

pp. 367-371 ◽

Cited By ~ 6

Author(s):

M. Tadi

Keyword(s):

Optimal Estimation ◽

Two Dimensional ◽

Zeroth Order ◽

First Order ◽

Conductivity Profile ◽

Boundary Measurements ◽

Conductivity Function ◽

The Given ◽

Close Estimate ◽

Dimensional Domain

This paper is concerned with an inverse problem for the conduction of heat in a two-dimensional domain. It seeks to recover the subsurface conductivity profile based on the measurements obtained at the boundary. The method considers a temporal interval for which time-dependent measurements are provided. It formulates an optimal estimation problem which seeks to minimize the error difference between the given data and the response from the system. It uses a combination of the zeroth-order and the first-order Tikhonov regularization to stabilize the inversion. The method leads to an iterative algorithm which, at every iteration, requires the solution to a two-point boundary value problem. A number of numerical results are presented which indicate that a close estimate of the thermal conductivity function can be obtained based on the boundary measurements only. [S0022-1481(00)00902-6]

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Lifted Inference with Tree Axioms

10.24963/kr.2021/57 ◽

2021 ◽

Author(s):

Timothy van Bremen ◽

Ondřej Kuželka

Keyword(s):

Binary Relation ◽

Domain Size ◽

Weighted Sum ◽

Order Model ◽

Lifted Inference ◽

Past Work ◽

First Order ◽

Graph Theoretic ◽

Model Counting ◽

The Given

We consider the problem of weighted first-order model counting (WFOMC): given a first-order sentence ϕ and domain size n ∈ ℕ, determine the weighted sum of models of ϕ over the domain {1, ..., n}. Past work has shown that any sentence using at most two logical variables admits an algorithm for WFOMC that runs in time polynomial in the given domain size (Van den Broeck 2011; Van den Broeck, Meert, and Darwiche 2014). In this paper, we extend this result to any two-variable sentence ϕ with the addition of a tree axiom, stating that some distinguished binary relation in ϕ forms a tree in the graph-theoretic sense.

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